Distributed formation shape control of multi-robot systems

Multi-robot system enjoys broad applications in civilian, industrial, and military areas, as team cooperation offers significant improvements of efficiency and system robustness. As an essential problem of multi-robot researches, the formation control problem that requires robots to form a desired formation shape has been intensively investigated in the past two decades. According to the type of constraints specifying the desired formation shape, most of the existing results can be classified as: 1) position-based; 2) displacement-based; 3) distance-based; and 4) bearing-based. However, few of these control laws can handle the formation shape control problem where the formation of the multi-robot system converges to the desired shape up to similar transformations (translation, rotation, scaling and reflection). The above motion freedoms enable us to maneuver the entire team by controlling a few agents called leaders when operating in unstructured environments. Moreover, the cost of sensors with different modalities and capabilities can vary significantly and so does the aggregate budget for large-scale multi-robot systems. Motivated by these issues, this thesis is dedicated to studying distributed formation shape control with full degrees of freedom by exploring the trade-off between requirements on sensor capabilities and sensing / communication graph topologies. As the first contribution of this thesis, a dynamic formation control problem for cooperative agents with discrete-time dynamics over directed graphs is proposed, where the desired formation varies with time and only its geometric shape is predefined. Matrix-valued Laplacian approach has been adopted to tackle this problem in the continuous-time setting. However, the discrete-time counterpart is more challenging due to the constraint in information exchange. On the other hand, observe that in many real operations, at a given time instant, the robots will be able to plan their target formation configurations for a period of time ahead. We propose preview-based P-like and PD-like controllers for the formation control. The controllers with proper parameter setting are proved to be effective to address the dynamic formation control problem. While the above-mentioned approach can achieve dynamic formation control with full degrees of freedom, it requires global position measurements, which are not available in environments without external positioning systems. Thus, we formulate a formation shape control problem where only relative position measurements in local coordinate frames are required. To address this problem, we develop a Ratio-of-Distance (RoD, the ratio of distances of a pair of edges joining a common vertex) rigidity theory to study when a framework can be uniquely determined by a set of RoD constraints up to similar transformations. Its relations to three existing rigidity theories (distance, bearing and angle rigidity theories) are established. The proposed RoD rigidity theory is further applied to the RoD-based similar formation stabilization problem where the desired formation shape is expressed as a set of RoD constraints. Notice that relative position measurements can only be obtained by data fusion of distance and bearing measurements and hence may not be of high accuracy due to different sampling times and frequencies. Hence, we consider a formation shape control problem where only bearing and RoD measurements in orient-aligned local coordinate frames are required. To solve this problem, a Bearing-Ratio-of-Distance (B-RoD, mixed bearing and RoD measurements) rigidity theory is developed to study when a framework can be uniquely determined by a set of B-RoD constraints up to directly similar transformation (translation, positive scaling). The proposed B-RoD rigidity theory is further applied to the B-RoD-based directly similar formation stabilization problem where the desired formation shape is expressed as a set of B-RoD constraints. Another method to relax the requirement of relative position measurements is to use distance and odometry measurements. In this direction, we study the problem of distance-based relative docking of a single robot and formation control of multi-robot systems. In particular, an integrated localization and navigation scheme is proposed for a robot to navigate itself to a desired relative position with respect to a fixed landmark at an unknown position, where only range and odometry measurements are used. By carefully embedding historical measurements into equilibrium conditions, we design an integrated estimation-control scheme to achieve the relative docking globally and asymptotically. It is rigorously proved that the robot will converge to the desired docking position asymptotically provided that control gains are chosen to satisfy certain conditions. This scheme is further extended to multi-robot systems to consider an integrated relative localization and formation control problem. Unlike widely used spatial cooperation in existing literature, we propose to exploit both spatial and temporal cooperation for achieving formation control. It is proved that multi-robot formation can be achieved with zero error for directed acyclic graphs (DAG). Based on the above-mentioned integrated scheme, we next consider achieving formation shape control with full degrees of freedom under the same measurements. To this end, we study the distance-based similar station keeping and similar formation control problems for a single robot and multi-robot systems, respectively. Specifically, the integrated scheme is adopted for a robot to navigate itself to the desired position, which forms a prescribed shape with fixed landmarks at unknown positions, based on range measurements with respect to landmarks and odometry measurements. A conflict of the persistent excitation (P.E.) existing in this scheme is resolved by introducing an autonomous system, whose output is regulated by a carefully designed function of distance measurements. It is shown that the robot will be steered to the desired position, with global asymptotic convergence if the control gains are properly selected. This scheme is further extended to consider an integrated relative localization and similar formation control problem of multi-robot systems. It is proved that multiple robots can be driven to a configuration, which is a similar transformation of a given template, with zero error for DAG.Doctor of Philosoph